Uniqueness of Quantum States Compatible with Given Measurement Results
We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a -dimensional Hilbert space, there exists a set of observables that uniquely determines any pure state. We show that for case (2), observables suffice to uniquely determine any pure state. Thus there is a gap between the results for (1) and (2), and we give some examples to illustrate this. Unique determination of a pure state by its reduced density matrices (RDMs), a special case of determination by observables, is also discussed. We improve the best known bound on local dimensions in which almost all pure states are uniquely determined by their RDMs for case (2). We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low rank quantum states.
pacs:03.65.Ud, 03.67.Mn, 89.70.Cf
I I. Introduction
In a -dimensional Hilbert space , the description of any quantum state generated by a source can be obtained by quantum tomography. For any density matrix , which is Hermitian and has trace , independent measurements are sufficient and necessary to uniquely specify . When is a pure state, one may not need as many measurements to uniquely determine . As we will see later, however, exactly what is meant by “uniquely” in this context needs to be specified.
Consider a set of linearly independent observables
where each is Hermitian. Measurements on state with respect to these observables give the following average values
We denote the set of these for all states as
For a pure state , these values are given by
and we denote the set of these values for all pure states as the joint numerical range
In this work we consider two different kinds of “unique determinedness” for :
We say is uniquely determined among pure states (UDP) by measuring if there does not exist any other pure state which has the same measurement results as those of when measuring .
We say is uniquely determined among all states (UDA) by measuring if there does not exist any other state, pure or mixed, which has the same measurement results as those of when measuring .
It is known that there exists a family of observables such that any pure state is UDP, in contrast to the observables in the general case of quantum tomography Heinosaari et al. (2011). The physical meaning for this case is clear: it is useful for the purpose of quantum tomography to have the prior knowledge that the state to be reconstructed is pure or nearly pure. Many other techniques for pure state tomography have been developed, and experiments have been performed to demonstrate the reduction of the number of measurements needed Weigert (1992); Amiet and Weigert (1999); Finkelstein (2004); Flammia et al. (2005); Gross et al. (2010); Cramer et al. (2010); Liu et al. (2012).
When the state is UDP, to make the tomography meaningful, one needs to make sure that the state is indeed pure. This is not in general practical, but one can readily generalize the above mentioned UDP results to low rank states, where the physical constraints (e.g., low temperature, locality of interaction) may ensure that the actual physical state (which ideally supposed to be pure) is indeed low rank. If the state is UDA, however, in terms of tomography one do not need to bother with these physical assumptions, because in the event there is only a unique state compatible with the measurement results, which turns out to be pure (or low rank).
There is also another clear physical meaning for the states that are UDA by measuring . Consider a Hamiltonian of the form
Then any unique ground state of is UDA by measuring . This is easy to verify: if there is any other state that gives the same measurement results, then has the same energy as that of , which is the ground state energy. Therefore, any pure state in the range of must also be a ground state, which contradicts the fact that is the unique ground state. In other words, UDA is a necessary condition for to be a unique ground state of . It is in general not sufficient, but the exceptions are likely rare Chen et al. (2012, 2012a).
The uniqueness properties for pure states, for both UDP and UDA, have also been studied extensively in the case of multipartite quantum systems, where the observables correspond to reduced density matrices (RDMs). That is, the observables are chosen to act nontrivially on only some subsystems. For an -particle system and a constant , there are a total of -RDMs, and the corresponding measurements are those -body operators. For example, for a three-qubit system and , one can choose as all the one and two-particle Pauli operators. Of course, one can also choose to look at some of the -RDMs, rather than all of them. For instance, for a three-particle system, one can look at -RDMs of particle pairs and .
It is known that almost all three-qubit pure states are UDA by their -RDMs Linden et al. (2002). These authors also show that UDP implies UDA for three-qubit pure states, for -RDMs. This result can be further improved to -RDMs of particle pairs and Chen et al. (2012). More generally one can consider a three-particle system of particles with Hilbert spaces whose dimensions are , respectively. If , then almost all pure states are UDA by their -RDMs of particle pairs and . In contrast, if , then almost all pure states are UDP by their -RDMs of particle pairs and , as shown by Diosi Diósi (2004).
For -particle quantum systems with equal dimensional subsystems, almost all pure states are UDA by their -RDMs of just over half of the parties (i.e., ). Furthermore, properly chosen RDMs among all the -RDMs suffice Jones and Linden (2005). W-type states are UDA by their -RDMs, and of those -RDMs are enough Parashar and Rana (2009). General symmetric Dicke states are UDA by their -RDMs Chen et al. (2012b). It has been shown that the only -particle pure states which cannot be UDP by their -RDMs are those GHZ-type states, and the result is further improved to the case of UDA Walck and Lyons (2008). Their results also show that UDP implies UDA for -qubit pure states, for -RDMs.
Despite these many results, there is no systematic study of these two different types of uniqueness for pure states. This will be the focus of this paper, where we are interested in knowing for given measurements , whether UDP and UDA are the same, or are different. We will give a general argument that there is a gap between the number of observables needed for the two different cases. However, in many interesting circumstances, they can coincide. Our discussions extend naturally to the case of low rank quantum states instead of just pure states. Here one can also look at two kinds of uniqueness when measuring given observables : one is uniqueness among all low rank states, the other is among all states of any rank.
We organize the paper as follows. In Sec. II, we first show that there is a set of observables that insures every pure state is UDA; which should be compared to the UDP result . Thus in general there is a gap between the optimal results for the UDP and UDA cases, and we illustrate this with some examples. Sec. III discusses the case of observables corresponding to RDMs of a multipartite quantum state, where for the three particle case, we show that if , then almost all pure states are UDA by their -RDMs of particle pairs and , improving the bounds given in Linden and Wootters (2002). However this still leaves a gap with the Diosi result for the case of UDP in Diósi (2004). We further discuss circumstances where UDP can imply UDA for all pure states. In Sec. IV, we show that when there are only two independent measurements performed, then UDP always implies UDA, by making use of convexity of the numerical range of operators. In a more general case, if there is a compact group of symmetries of the state space which has the span of the operators measured as its set of fixed points, then UDP implies UDA for all pure states. We analyze the possible dimensions for those fixed point sets. A summary and some discussions are included in Sec. VI.
Ii II. The number of observables for UDA
In this section, we discuss the minimum number of observables needed to have all pure states be UDA. We start by choosing a Hermitian basis for the operators on . Without loss of generality we choose , the identity operator on , which has trace . We further require that the ’s are orthogonal, in the sense that for ,
The Hermitian matrices form a real inner product space with inner product , so such a basis exists for any dimension . For instance, for the qubit case (), we can choose the Pauli basis
For the qutrit case (), one can choose for , where s are the Gell-Mann matrices given by
For general , one can choose for , where s are the generalized Gell-Man matrices.
We can now write any density operator as
where , and where has real entries.
We have , therefore , and the equality holds if is a pure state. However, not every state satisfying is a pure state. Indeed, is a pure state if and only if , which gives equations that needs to satisfy.
If one of the observables is a multiple of the identity, then we can drop it from the list of observables without affecting UDA and UDP. If two states agree on an observable , then they agree on for any real scalar , so we can adjust each of the observables to have trace zero without affecting UDA or UDP. Hence hereafter we assume all are traceless.
For any observable , we can expand in terms of as
Then the average value of is given by
To discuss the problem for any pure state to be UDA, the constant and constant factor can be ignored, as these are the same constants for all states. Therefore we have
where means that the average value of for the state is geometrically equivalent to the projection of onto .
Alternatively, define by . Let be the linear subspace of spanned by , and let be the orthogonal projection from onto . Then and have the same kernel, namely . Thus for states , we have if and only if , so in considering UDA and UDP we can treat as being the orthogonal projection onto .
If we subtract the density matrix from all states, then the translated set of states sits in the real dimensional subspace of trace zero Hermitian matrices. In this sense, we are actually working with real geometry in . All quantum states then sit inside the -dimensional unit ball, with pure states corresponding to unit vectors, but not every vector on the unit -dimensional sphere is a pure state. The observables span an -dimensional subspace that all the quantum states will be projected onto. We will simply say the subspace is spanned by when no confusion arises, and we will no longer distinguish an operator from the corresponding vector . Indeed we only consider the real span of , and we denote it by . For each , there is an orthogonal subspace in of dimension , which we denote by . Here we are taking the orthogonal complement in the space of traceless Hermitian matrices, so that every is traceless.
We now are ready to state our first theorem.
For a -dimensional system (), there exists a set of observables for which every pure state is UDA.
To see why this is the case, note that in the above-mentioned geometrical picture, it is clear that a pure state is UDA by measuring if there does not exist any operator , such that is positive. One sufficient condition will then be that any operator has at least two positive and two negative eigenvalues. We will use this sufficient condition to construct a desired .
In order to construct , we provide a set of linearly independent Hermitian matrices explicitly, such that the Hermitian matrix
has at least two positive eigenvalues for any nonzero real vector .
Our construction is motivated by and similar to the diagonal filling technique used in Ref. Cubitt et al. (2008), but along the other direction of the diagonals.
This then means that measuring observables is enough for any pure state to be UDA, which proves the theorem. There are indeed technical details to be clarified that we leave to Appendix A.
If we compare our results with those given in Heinosaari et al. (2011), which shows that measuring observables are enough for any pure state to be UDP, there exists an obvious gap. We claim that this gap indeed cannot be closed in general. To see this, let us look at the simplest case of , where the results just compared state that observables are enough for any pure state to be UDP but observables are enough for any pure state to be UDA.
If one can measure a particular set with observables and have all pure states be UDA, then also every state also must be UDP for measuring . According to Heinosaari et al. (2011), this only happens if contains a single invertible traceless operator , meaning is rank . Without loss of generality we can assume the largest eigenvalue to be positive with an eigenstate . Then is not UDA by measuring since as observed in Heinosaari et al. (2011) there exists a mixed state which also has the same average values as those of . Therefore, one cannot only measure observables for all pure states to be UDA.
For general , our construction needs observables. We do not know whether this is the optimal construction, but it is very unlikely one can get this down to . In other words, in general UDA and UDP for pure states should be indeed two different concepts and there should always be gaps between the number of observables needed to be measured for each case to uniquely determine any pure quantum state. This is one exception though, which is for the qubit case (i.e., ) where it is shown in Heinosaari et al. (2011) that for all pure states to be UDP, one needs to measure variables, which then uniquely determine any quantum state among all states.
Finally, we remark that our results in Theorem 1 naturally extend to the case of low rank states. That is, for a rank quantum state , we can similarly consider two different cases: (1) is uniquely determined by measuring among all rank states (which was considered in Heinosaari et al. (2011)) (2) is uniquely determined by measuring among all quantum states of any rank.
For a -dimensional system () measuring observables is enough for a rank state to be uniquely determined among all states.
Iii III. The case of reduced density matrices
In this section we discuss the case where the Hilbert space is a multipartite quantum system, where the observables correspond to the reduced density matrices (RDMs). That is, the observables are chosen to be acting nontrivially only on some subsystems. For instance, for a three-qubit system, the observables corresponding to the -RDMs of particle pairs can be chosen as
where are Pauli operators acting on the th qubit.
For simplicity in this section we consider only -particle systems, labeled by , and each with Hilbert space dimension , respectively. That is, and . Nevertheless, our method naturally extends to systems of more than -particles.
Recall that for a three particle system, it is known that almost all three-qubit pure states are UDA by their -RDMs Linden et al. (2002). This result can be further improved to -RDMs of particle pairs and Chen et al. (2012). More generally, if , then almost all pure states are UDA by their -RDMs of particle pairs and Linden and Wootters (2002). In contrast, if , then almost every pure state is UDP by its -RDMs of particle pairs and Diósi (2004).
We notice that different from the discussion in Sec. II, one no longer considers uniqueness for all pure states, but ‘almost all’ of them. This means there exists a measure zero set of pure states which are not uniquely determined. For instance, for the three qubit case, any state which is local unitarily equivalent to the GHZ type state
cannot be UDP, as any state of the form has the same -RDMs as those of . This means that, for a three qubit pure state , it is either UDA, or not UDP. In other words, if any three qubit pure state is UDP, then it is UDA by its -RDMs of particle pairs and . In this sense, we say in this case UDP implies UDA for all pure states.
However, for the general case of a three particle system, there is a gap between known results of UDA and UDP. Our following result improves the bound for the UDA case.
If , then almost every tripartite quantum state is UDA by its -RDMs of particle pairs and ,
To see why this is the case, an arbitrary pure state of this system can be written as
If there is another state which agrees with in its subsystems and , then we can find a pure state which agrees with on the subsystem and also agrees with in subsystems and .
Since the rank of the -RDM of the subsystem is at most , the pure state can be written as a superposition of as follows.
for any . Here will be vectors (perhaps unnormalized) in .
The states can be chosen to be orthonormal vectors in the subsystem , and then for almost all states , the set of will be linearly independent. Let us write . For any , we will have
Now let’s consider the subsystem . Since and have the same RDMs for particles , this gives
Now let us define . Then Eq. (21) is a linear equation system with variables . It is not hard to verify that
is a solution to the equation system, which corresponds to the state .
Now we need to show that when , Eq. (21) has only one solution which is given by Eq. (22). It turns out that this is indeed the case which then proves Theorem 3. In fact, the linear equations above are generically linearly independent. To see this, let’s fix and , the right-hand side of Eq. (21) is where . Then the coefficient matrix can be written as the following:
The entry in the above matrix is .
If there are more than solutions, then the determinant of the above matrix should be zero. Note that the determinant can be written as a polynomial of ’s and ’s. Since appears only once in the polynomial, the determinant of the top by submatrix must be non-zero generically. Therefore, linear equations are sufficient to determine variables.
However, we do not know whether the sufficient condition given by Theorem 3 for almost all three-particle pure state to be UDA by its -RDMs of particle pairs and is also necessary. This still leaves a gap between the result of Theorem 3 for UDA, and the result for UDP in Diósi (2004). They both only coincide when , i.e., the three qubit case. It remains open for other cases, whether UDP can imply UDA.
Following a similar discussion as in Sec. II, our result in this section also extends to uniqueness of low rank quantum states. In particular, we have the following theorem.
Almost every tripartite density operator acting on the Hilbert space with rank no more than can be uniquely determined among all states by its -RDMs of particle pairs and .
This result is to our knowledge, the first one for uniqueness of mixed states with respect to RDMs. The proof is a direct extension of that for Theorem 3, but with more lengthy details that we will include in Appendix B.
Let us look at some consequences of Theorem 4. Consider a four qubit system with qubits , and look at the qubits as a single systems . Then Theorem 4 says also that almost all four qubit states of rank are UDA by their RDMs of particles and , or one can say that almost all four qubit states of rank are UDA by their -RDMs. This is indeed consistent with the multipartite result in Jones and Linden (2005) which states that almost all four-qubit pure states are UDA by their -RDMs, and our result is indeed stronger. This demonstrates that our analysis naturally extends to systems of more than -particles. We also remark that the rank of a state which could be UDA by its -RDMs needs to be relatively low, otherwise one can always find another state with lower rank which has the same -RDMs as those of Chen et al. (2012c).
Iv IV. The case of only two observables
In Sec. II and Sec. III, we discussed the difference and coincidence between the two kinds of uniqueness for pure states, UDA and UDP, which in general are not the same thing. However, in certain interesting circumstances such as the three qubit case with respect to -RDMs, and in general the -qubit case respect to -RDMs, they do coincide. Starting from this section we would like to build some general understanding of the circumstances when UDP implies UDA for all pure states.
We start from the simplest case of , where only two observables are measured, i.e., . Intuitively, in this extreme case almost no pure state can be uniquely determined, either UDA or even UDP. However there are also exceptions. For instance, if one of the observables, say , has a nondegenerate ground state , then is UDA (hence, of course, UDP) even by measuring only. One would hope this is the only exception, that is, for a pure state , either it is UDA, or it is not UDP, when only two observables are measured. We make this intuition rigorous by the following theorem.
When only two observables are measured, i.e., , UDP implies UDA for any pure state , regardless of the dimension .
To prove this theorem, recall that measuring (i.e., measuring every observable in ) for all quantum states returns the set given by Eq. (3). We know that is a convex set, meaning for any , we have for any .
For pure states, the corresponding set of average values is given by as defined in Eq. (5). Unlike , in general is not convex. Nevertheless, it is easy to see that when is convex.
For , the Hausdorff–Toeplitz theorem Toeplitz (1918); Hausdorff (1919) gives convexity of the numerical range of any operator, which in turn shows that is convex. We explain it briefly here. For any operator acting on a Hilbert space , the numerical range of is the set of all complex numbers , where ranges over all pure states in .
Note that one can always write as
If we define and then clearly both and are Hermitian. Then is nothing but the numerical range of and hence is convex.
Furthermore, by studying the properties of the numerical range, it was shown in Embry (1970) (using different terminology) that if a pure state is UDP, the point must be an extreme point of . Here is an extreme point of the convex set if there do not exist , such that for some .
Because , is also an extreme point of . One can further show that for any extreme point of , and any quantum state with , any pure quantum state in the range of will also have . This then implies that if a pure state is UDP by measuring , it must also be UDA, which proves the theorem.
Again, all the technical details of the proof will be presented in Appendix C.
In an attempt to extend Theorem 5 to the case, a natural question that one could ask is whether or not UDP implies UDA whenever is convex. Unfortunately this is not the case, as demonstrated by the following example.
For the qutrit case (), consider the observables , where the s are the Gell-Mann matrices given in Eq. (9). These are the Pauli operators embedded in the qutrit space. It is easily verified that in this case, is the Bloch sphere together with its interior and is thus convex. Nonetheless, the unique pure state compatible with measurement result is the state , even though there are many mixed states sharing this measurement result, such as .
Therefore, although the Hausdorff–Toeplitz theorem Toeplitz (1918); Hausdorff (1919) is famous for showing the convexity of numerical range of any operator, there is indeed a deeper reason than just the convexity of the numerical range which governs the validity of Theorem 5. We leave the more detailed discussion to Appendix C.
V V. Symmetry of the state space
In this section, we discuss some circumstances where UDP implies UDA in a more general context where more than two observables are measured, i.e., . Our focus is on the symmetry of the set of all quantum states. For a -dimensional Hilbert space we denote this set of states by , that is
Note that is convex, as we know that for any , for all . Furthermore, the extreme points of are all the pure states. is also called the state space for all the operators acting on .
We now explain the intuition. If has a certain symmetry, then two pure states and that are ‘connected’ by the symmetry will give the same measurement results, and states fixed by the symmetry will also be fixed by the projection onto the space of observables. In this situation, UDP implies UDA for all pure states.
To make this intuition concrete, let us first consider an example for , i.e., the qubit case. We know that can be parameterized as in Eq. (10), where for , are chosen as Pauli matrices given in Eq. (8). Here is the Bloch ball as shown in FIG. 1. The Bloch ball is clearly a convex set and the extreme points are those pure states on the boundary, which give the Bloch sphere.
We know that geometrically, measuring the observables in corresponds to the projection onto the plane spanned by . For example, if we measure the Pauli and operators, then geometrically this corresponds to the projection of the Bloch ball onto the plane. Since the Bloch ball has reflection symmetry with respect to the plane, two pure states (e.g. points and ) connected by that symmetry will project onto the same measurement result , as will all mixtures of and . Hence neither UDP nor UDA hold for such pure states for measuring and . On the other hand, pure states fixed by the reflection symmetry are also fixed by the projection onto the plane. These are precisely the points on the Bloch sphere that are in the plane (e.g. the points and in FIG. 1), and for such pure states both UDP and UDA hold. Therefore, for the observables we conclude that UDP = UDA.
Now let us look at another case where we only measure the Pauli operator. Consider the group of symmetries of the Bloch ball consisting of rotation around the axis. (Rotation by angle , is shown in FIG. 1. In that figure, point will become point after this particular rotation, and indeed both points and yield the same measurement result, which is represented by point on the axis.) Note that two points on the Bloch sphere will project to the same measurement result on the axis if and only if they are in the same orbit under the rotation group. Thus a measurement result will come from a single pure state exactly when that pure state is a fixed point, and hence either both or neither of UDP and UDA hold for each pure state. For example, the point is fixed by the rotation, and is uniquely determined by the measurement of among all states. corresponds to the eigenstate of the Pauli operator. Therefore, the rotational symmetry of the Bloch ball along the axis gives UDP =UDA for any pure state when measuring the Pauli operator, which corresponds to the axis.
Mathematically, a symmetry of is an affine automorphism of . If is unitary, the map taking to is such an affine automorphism (which for will just be rotation around some axis of the Bloch ball). For instance, the rotation symmetry along the axis by an angle is given by conjugation by the unitary operator . If is the conjugate linear map given by complex conjugation in the computational basis (), then the map taking to is the transpose map. For , this map is reflection of the Bloch ball in the xy-plane.
Recall that for a set of observables , we denote the real linear span by . When discussing the uniqueness problems, it makes no difference if we append the identity operator to . Let us then assume . We are now ready to put our intuition into a theorem.
Assume there exists a compact group of affine automorphisms of whose fixed point set is . Then each pure state acting on which is UDP for measuring is also UDA.
In the first example above, the group for the reflection consists of the two element group generated by the reflection. In the rotation example, we can take the group to consist of all rotations around the given axis. We will leave the detailed mathematical proof of Theorem 6 to Appendix D, where operator algebras are one ingredient of the proof.
To motivate some further consequences of Theorem 6, consider a simple example. If consists of a basis of diagonal matrices (i.e., a set of mutually commuting observables), then for any pure state, UDP implies UDA by Theorem 6. Here the group of symmetries can be taken to be conjugation by all diagonal unitaries. This group has fixed point set . In a more general case, if the complex span of is a *-subalgebra of the operators acting on , then UDP = UDA for all pure states for measuring . This is a natural corollary of Theorem 6 that we will also discuss in detail in Appendix D.
Vi VI. Conclusion and Discussion
In this work, we have discussed the uniqueness of quantum states compatible with given results for measuring a set of observables. For a given pure state, we consider two different types of uniqueness, UDP and UDA. We have taken the first step to study their relationship systematically. In doing so we have established a number of results, but also leave with many open questions.
First of all, although in general UDP and UDA are evidently different concepts, their difference is surprisingly ‘not that large’. Specifically in the sense of general counting of the number of variables one needs to measure to uniquely determine all pure states in a dimensional Hilbert space. Compared to full quantum tomography which requires variables measured to uniquely determine any quantum state, the observables we have constructed to uniquely determine any pure state among all states is a significant improvement. It is indeed larger than the observables given in Heinosaari et al. (2011) to uniquely determine any pure state among all pure states, but the difference is only linear in . We do not know whether there could be another construction for which we could further close the linear difference between UDA and UDP, to leave only a constant gap for large .
When the Hilbert space is a multipartite quantum system, and the observables correspond to the RDMs, we focused on the situation when ‘almost all pure states’ are uniquely determined. We considered a -particle system with Hilbert space , and showed that if , then almost all pure states are UDA by their -RDMs of particle pairs and . This improves the results of Linden and Wootters (2002), where is required; however it still leaves a gap compared to the Diosi UDP result which states that for , almost all pure states are UDP by their -RDMs of particle pairs and . Because our proof only gives a sufficient condition for UDA, we do not know whether it can be further improved. We also do not have an example showing there is indeed gap between UDA and UDP for almost all three-particle pure states to be uniquely determined by -RDMs of particle pairs and .
Finally, we considered situations for which we can show that UDP implies UDA. These include: (i) the general -qubit system; (ii) the -qubit system when we consider uniqueness for almost all pure states and the measurements corresponds to -RDMs; (iii) when only two observables are measured; and (iv) the observables measured correspond to some symmetry of the state space. However we do not know how far we are from enumerating all the possible situations that UDP implies UDA, when considering uniqueness for all pure states or almost all pure states. In principle one can even consider the relationship between UDP and UDA for special subsets of pure states.
We believe our systematic study of the uniqueness of quantum states compatible with given measurement results shed light on several aspects of quantum information theory and its connection to different topics in mathematics. These include quantum tomography and the space of Hermitian operators, unique ground states of local Hamiltonians and general solutions to certain linear equations, measurements and numerical ranges of operators, and the geometric meaning of measurements and the symmetry of state space. We thus conclude with several open questions that we believe warrant further investigation.
Theorem 1 can be implied by the following Lemma.
There exists a set of linearly independent Hermitian matrices , such that the Hermitian matrix
has at least two positive eigenvalues for any nonzero real vector .
We prove the statement by giving an explicit construction. Our proof is motivated by and similar to the diagonal filling technique used in Ref. Cubitt et al. (2008), but along the other direction of the diagonals.
We will need the lemma 9 from Ref. Cubitt et al. (2008) about totally non-singular matrix, which we restate as Lemma 3 in the following. For simplicity, we also assume that the totally non-singular matrix is real. Therefore, for any length and , there is linearly independent real vectors such that every nonzero linear combination of them has at least nonzero entries.
Let be a matrix. We will always fix the diagonal to be zero, namely for . In the upper triangular part of the matrix not including the diagonal, there are lines of entries parallel to the antidiagonal. That is, each line contains entries with and where goes from to . We will call it the -th line of the matrix in the following. We also call the set of entries with the -th antidiagonal. It is easy to see that the length of the -th line is
So the length for , and we can find real vectors for which every nonzero linear combination has at least nonzero entries. For each of the vectors, we can form two Hermitian matrices. One of them is the symmetric one whose -th line is filled with the vector, and the lower triangular part determined by the Hermitian condition. Such a matrix is a real symmetric matrix having nonzero entries only on the -th antidiagonal. We will call it a real -th line matrix. The other is the one with -th line filled with the vector multiplied by , and lower part is determined by the Hermitian condition. This is a matrix consisting of purely imaginary entries on the -th antidiagonal and we call it an imaginary -th line matrix.
Now we prove that the constructed matrices satisfy our requirement. First we prove that the matrices are linearly independent. It suffices to show that the matrices of nonzero -th line is linearly independent. Let be the set of linearly independent real vectors chosen for the -th line. We need to show that is linearly independent over . If the contrary is true, that is, there exists complex numbers not all zero such that
This is equivalent to
From the above two equations, we get and which is a contradiction.
Next, we prove that for any nonzero real coefficient , the matrix has at least two positive eigenvalues. Let be the largest such that there is a -th line matrix whose coefficient is nonzero. Then, either the real -th line matrices or the imaginary ones have nonzero coefficients. By the construction, this implies that there is at least two nonzero entries on the th line of the matrix . Let the nonzero entries be . We then have a principle submatrix of that has the form
where are two unknown number and represents the complex conjugate of . This matrix has trace and determinant . Therefore, it has exactly two positive eigenvalues. As it is a principle submatrix of matrix , follows from Theorem 7, has at least two positive eigenvalues.
The number of matrices thus constructed is the summation
which can be computed to be
Discussion: We note that our construction will also imply that the matrix has at least two negative eigenvalues, thus at least rank . But our bound is even better than the bound on the dimension of subspaces in which every matrix has rank . This is not a contradiction as we are considering all real combinations. For example, the case of has two matrices for our purpose, namely
These two matrices do satisfy our requirements, but their span over contains a rank matrix .
Generalization: Similarly, length for , and we can find real vectors for which every nonzero linear combination has at least nonzero entries. For each of the vectors, we can also form two Hermitian matrices. Such constructed matrices are linearly independent and any real linear combination has at least positive eigenvalues.
There exists a set of linearly independent Hermitian matrices , such that the Hermitian matrix
has at least positive eigenvalues for any nonzero real vector .
We just follow the lines of the proof of Lemma 1. To complete our argument, we need to show that any by invertible traceless, Hermitian, upper left triangler matrix has exactly positive eigenvalues.
Let’s prove this claim by induction. When , it is already known. Let’s assume this claim holds true for any . Then for , we can write such matrix in the following form
One may observe that, by deleting the first and the last rows/columns, we will have a by invertible, traceless, Hermitian, upper left triangler submatrix.
From our assumption, this submatrix has exactly positive eigenvalues which means has at least positive eigenvalues and at least negative eigenvalues.
Note that its determinant equals to . This follows that has exactly positive eigenvalues which completes our argument.
The number of matrices thus constructed is the summation